How to use diagnostic tests.


BaalChatzaf

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Consider Sick City, USA with a population of 1,000,000. A survey has shown 10,000 of the 1,000,000,

(that is 1 %) has cancer of the little toe. The best diagnostic test ToeTest © gives a 95% positive result when used on people with cancer of the little toe. ToeTest give a 5% false positive rate, i.e. the test says the patient has no cancer of the little toe, when in fact he does.

Now here is the problem. A random person from Sick City, USA comes into the doctor's office and the doctor administers ToeTest © and gets a positive result. Question. Given that ToeTest the diagnostic test says the patient has cancer of the little toe, what is the probability that the patient really has cancer of the little toe, given that ToeTest registers positive?

Is the answer:

a) .84

b) .65

c) .16

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Consider Sick City, USA with a population of 1,000,000. A survey has shown 10,000 of the 1,000,000,

(that is 1 %) has cancer of the little toe. The best diagnostic test ToeTest © gives a 95% positive result when used on people with cancer of the little toe. ToeTest give a 5% false positive rate, i.e. the test says the patient has no cancer of the little toe, when in fact he does.

Now here is the problem. A random person from Sick City, USA comes into the doctor's office and the doctor administers ToeTest © and gets a positive result. Question. Given that ToeTest the diagnostic test says the patient has cancer of the little toe, what is the probability that the patient really has cancer of the little toe, given that ToeTest registers positive?

Is the answer:

a) .84

b) .65

c) .16

with cancer: 95% positive, 5% negative

without cancer: Unknown. I will assume 0% positive, 100% negative

Then the probability that the patient has cancer of the little toe is 100%.

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ToeTest give a 5% false positive rate, i.e. the test says the patient has no cancer of the little toe, when in fact he does.

I first answered c) .16.

Then I saw the problem is ill-formed. "False positives" means "those that test positive (T+) but do not have the disease (D−)" (link).

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ToeTest give a 5% false positive rate, i.e. the test says the patient has no cancer of the little toe, when in fact he does.

I first answered c) .16.

Then I saw the problem is ill-formed. "False positives" means "those that test positive (T+) but do not have the disease (D−)" (link).

.16 is correct. Use Bayes Theorem. Five percent of the sick people who have ToeTest administered are told that the result is negative.

I am in the process of becoming a Born Again Bayesian .

There is a delightful non-technical book on Bayesian Statistics and decision making. Do read

The Theory that would not Die by Sharon Bertsch McGayne.

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Assume the entire population of 1000000 people is tested. Given your conditions:

Positive test results:
Well population = (1000000-10000) x .05 = 49500
Sick population = (10000) x .95 = 9500
Total positive results: 49500 + 9500 = 59000
Odds of a positive result being in the sick population = 9500/59000 = .161 (16%).

Given the simplicity of this problem, how would Bayes' theorem help? How you would use it to solve it more directly?

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Assume the entire population of 1000000 people is tested. Given your conditions:

Positive test results:

Well population = (1000000-10000) x .05 = 49500

Sick population = (10000) x .95 = 9500

Total positive results: 49500 + 9500 = 59000

Odds of a positive result being in the sick population = 9500/59000 = .161 (16%).

Given the simplicity of this problem, how would Bayes' theorem help? How you would use it to solve it more directly?

I stated the premises incorrectly. ToeTest gives a correct positive on a sick person with probability .95. When used on a well person it gives a false positive with probability .05.

P(x is sick | x tested +) =

P(x tested + | x is sick)*P(x is sick) / (P(x tested + | x is sick)* P(x is sick) + P(x tested + | x is not sick)* P(x is not sick))

which when you plug in the numbers comes out to .16.

Many of the physicians given this test come up with the probability .95 that the patient is sick because he tested positive.

It is no wonder that many doctors over prescribe antibiotics. They assume the worst case without a sufficient statistical basis.

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Yes. I quit a doctor I'd been going to for 15 years. He told me I shouldn't look stuff up on the internet and wrote "refused treatment" in my record because I wouldn't take a prescription he wrote for me. I found a doctor who believes in preventative medicine, nutrition and lifestyle and doesn't mind that I try to be as informed as possible about any conditions I have and even encourages me to send her the links. Night and day. Except for medical researchers I doubt the majority of doctors have particularly good critical reasoning skills but were good rote learners and authoritarians. Sad to say. There are, thankfully, exceptions if you can find them.

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